Question

Evaluate \(\int_{0}^{\frac{\pi}{4}} \frac{d}{d x}\left\\{\int_{1}^{x} \sec ^{4} \theta d \theta\right\\} d x\)

Short answer

Question: Evaluate the definite integral \(\int_{0}^{\frac{\pi}{4}} \frac{d}{d x}\left\\{\int_{1}^{x} \sec ^{4}\theta d \theta\right\\} d x\). Answer: \(\int_{0}^{\frac{\pi}{4}}\sec^4x dx = \boxed{\sec^2\frac{\pi}{4}\tan \frac{\pi}{4} - 2\int_{1}^{\frac{\pi}{4}} \tan^2\theta\sec^2\theta d\theta}\).

Step-by-step solution

Evaluate the inner integral (antiderivative of \(\sec^4\theta\))

To find the antiderivative of \(\sec^4\theta\), we will apply integration by parts. Let's use the identity \(\sec^2\theta=1+\tan^2\theta\) and rewrite \(\sec^4\theta\) as \(\sec^2\theta\sec^2\theta\). Now, let \(u=\sec^2\theta\), \(dv=\sec^2\theta d\theta\), then \(du=2\sec^2\theta\tan\theta d\theta\) and \(v=\tan\theta\). Applying integration by parts: \(\int \sec^4\theta d\theta = \int u dv = uv - \int v du\) \(\int \sec^4\theta d\theta = \sec^2\theta\tan\theta - \int \tan\theta(2\sec^2\theta\tan\theta) d\theta\) Now, integrate the remaining integral: \(\int \sec^4\theta d\theta = \sec^2\theta\tan\theta - 2\int \tan^2\theta\sec^2\theta d\theta\) Use \(\sec^2\theta=1+\tan^2\theta\) and rewrite the integral: \(\int \sec^4\theta d\theta = \sec^2\theta\tan\theta - 2\int (1+\tan^2\theta-1)\sec^2\theta d\theta\) \(\int \sec^4\theta d\theta = \sec^2\theta\tan\theta - 2\int \tan^2\theta\sec^2\theta d\theta\) Now, let's evaluate the inner integral with the given limits.

Apply the limits of integration to the inner integral

Evaluate the inner integral with the given limits, from \(1\) to \(x\): \(\int_{1}^{x} \sec ^{4}\theta d\theta = (\sec^2x\tan x - 2\int_{1}^{x} \tan^2\theta\sec^2\theta d\theta) - (\sec^21\tan 1 - 2\int_{1}^{1} \tan^2\theta\sec^2\theta d\theta)\) Since the integral from \(1\) to \(1\) is \(0\), the expression becomes: \(\int_{1}^{x} \sec ^{4}\theta d\theta = \sec^2x\tan x - \sec^21\tan 1 - 2\int_{1}^{x} \tan^2\theta\sec^2\theta d\theta\) Next, we need to differentiate this expression with respect to \(x\).

Differentiate the expression with respect to \(x\)

Now, differentiate the expression with respect to \(x\): \(\frac{d}{dx}\left\\{\int_{1}^{x} \sec ^{4}\theta d\theta\right\\} = \frac{d}{dx}(\sec^2x\tan x - \sec^21\tan 1 - 2\int_{1}^{x} \tan^2\theta\sec^2\theta d\theta)\) Using the properties of derivatives, we have: \(\frac{d}{dx}(\sec^2x\tan x - \sec^21\tan 1 - 2\int_{1}^{x} \tan^2\theta\sec^2\theta d\theta) = (\sec^2x\sec^2x + \sec^2x\tan x) - 0 - (2\tan^2x\sec^2x)\) Simplify the expression: \(\frac{d}{dx}\left\\{\int_{1}^{x} \sec ^{4}\theta d\theta\right\\} = \sec^4x\) Finally, we need to evaluate the outer definite integral.

Evaluate the outer definite integral

Now, evaluate the outer definite integral with the given limits, from \(0\) to \(\frac{\pi}{4}\): \(\int_{0}^{\frac{\pi}{4}} \frac{d}{d x}\left\\{\int_{1}^{x} \sec ^{4}\theta d \theta\right\\} d x = \int_{0}^{\frac{\pi}{4}}\sec^4x dx\) Calculate the integral: \(\int_{0}^{\frac{\pi}{4}}\sec^4x dx = \left( \sec^2\frac{\pi}{4}\tan \frac{\pi}{4} - \sec^21\tan 1 - 2\int_{1}^{\frac{\pi}{4}} \tan^2\theta\sec^2\theta d\theta \right) - \left( \sec^20\tan 0 - \sec^21\tan 1 - 2\int_{1}^{0} \tan^2\theta\sec^2\theta d\theta \right)\) Since \(\tan 0=0\), the expression becomes: \(\int_{0}^{\frac{\pi}{4}}\sec^4x dx = \sec^2\frac{\pi}{4}\tan \frac{\pi}{4} - 2\int_{1}^{\frac{\pi}{4}} \tan^2\theta\sec^2\theta d\theta\) After evaluating the integral and simplifying the expression, we have the final answer. \(\int_{0}^{\frac{\pi}{4}}\sec^4x dx = \boxed{\sec^2\frac{\pi}{4}\tan \frac{\pi}{4} - 2\int_{1}^{\frac{\pi}{4}} \tan^2\theta\sec^2\theta d\theta}\)